1. Combine teriyaki marinade and chicken breasts in a medium bowl or re-sealable plastic bag. Marinate chicken for 3-4 hours. 2. Prepare the dressing by combining all of the ingredients in a small saucepan over medium heat. Bring mixture to a rolling boil while stirring often with a whisk, then remove the pan from the heat to cool. When dressing has cooled, pour it into a covered container and chill. 3. When chicken breasts have marinated, preheat barbecue grill to high heat. Grill chicken for 3-4 minutes per side, or until done. 4. Combine the romaine and iceberg lettuce, red and green cabbage and 1 cup of shredded carrots in a large bowl with the dressing. Toss well. 5. Divide the tossed greens among four plates. Sprinkle 1/4 cup of green onions over each salad, followed by 1/3 cup of crispy chow mein noodles. 6. When the chicken breasts are done, slice each one, widthwise, into bite-size pieces. Sprinkle the sliced chicken breasts over each salad. 7. Place a 1/4 cup pile of shredded carrots in the center of each salad.